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9.E: Spin Angular Momentum (Exercises)

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    1. Find the Pauli representations of \(S_x\), \(S_y\), and \(S_z\) for a spin-1 particle.
    2. Find the Pauli representations of the normalized eigenstates of \(S_x\) and \(S_y\) for a spin-\(1/2\) particle.
    3. Suppose that a spin-\(1/2\) particle has a spin vector that lies in the \(x\)-\(z\) plane, making an angle \(\theta\) with the \(z\)-axis. Demonstrate that a measurement of \(S_z\) yields \(\hbar/2\) with probability \(\cos^2(\theta/2)\), and \(-\hbar/2\) with probability \(\sin^2(\theta/2)\).
    4. An electron is in the spin-state \[\chi = A\,\left(\begin{array}{c}1-2\,{\rm i}\\2\end{array}\right)\] in the Pauli representation. Determine the constant \(A\) by normalizing \(\chi\). If a measurement of \(S_z\) is made, what values will be obtained, and with what probabilities? What is the expectation value of \(S_z\)? Repeat the previous calculations for \(S_x\) and \(S_y\).
    5. Consider a spin-\(1/2\) system represented by the normalized spinor \[\chi =\left(\begin{array}{c}\cos\alpha\\\sin\alpha\,\exp(\,{\rm i}\,\beta)\end{array}\right)\] in the Pauli representation, where \(\alpha\) and \(\beta\) are real. What is the probability that a measurement of \(S_y\) yields \(-\hbar/2\)?
    6. An electron is at rest in an oscillating magnetic field \[{\bf B} = B_0\,\cos(\omega\,t)\,{\bf e}_z,\] where \(B_0\) and \(\omega\) are real positive constants.
      1. Find the Hamiltonian of the system.
      2. If the electron starts in the spin-up state with respect to the \(x\)-axis, determine the spinor \(\chi(t)\) which represents the state of the system in the Pauli representation at all subsequent times.
      3. Find the probability that a measurement of \(S_x\) yields the result \(-\hbar/2\) as a function of time.
      4. What is the minimum value of \(B_0\) required to force a complete flip in \(S_x\)?

    Contributors and Attributions

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)